In this letter, based on our recent work [1]}, we will discuss the
generation of the BAU via the leptogenesis mechanism in the framework of a
SUSY \(SU(5)\)  model extended by a \(D_{4}\times U(1)\)  symmetry. The dihedral
group \(D_{4}\)  is introduced to adress the neutrino flavor structure while \(U(1)\)  controls the invariance of the superpotential of the model. We present
here an update of the study done in Ref. [1]} where we assume that the
RH neutrino mass spectrum is not strongly hierarchical which requires taking
into account the contribution of the decay of the three RHNs. In
this case, we find that the \(CP\)  asymmetries for each Majorana neutrino \(\varepsilon _{N_{i}}\)  is dominated by the contribution of a next-to-leading
(NLO) order correction to the usual type I seesaw terms. In particular, we
find that the correlation between the baryon asymmetry parameter \(Y_{B}\) 
and the parameters from the NLO term satisfies the experimental bound of \(Y_{B}\)  from the Planck collaboration [3]}. We find also that the
predicted values of the effective Majorana mass \(m_{\beta \beta }\)  measured
in neutrinoless double beta decay (\(0\nu \beta \beta \) ) are testable at
future decay experiments.
